Answer
133 free throws, 193 2-point field goals, 43 3-point field goals
Work Step by Step
648 points
number of 2 point field goals is 22 fewer than five times number of 3 point field goals
number 1 point free throws is 60 fewer than number of 2 point field goals
$x$, $y$, $z$ is number of 1 point free throws, 2 point field goals, and 3 point field goals, respectively
$x+2y+3z=648$
$x=y-60$
$y=5z-22$
$x+2y+3z=648$
$(y-60)+2y+3z=648$
$y-60+2y+3z=648$
$3y+3z-60=648$
$3y+3z-60+60=648+60$
$3y+3z=708$
$(3y+3z)/3=708/3$
$y+z=236$
$(5z-22)+z=236$
$5z-22+z=236$
$6z-22=236$
$6z-22+22=236+22$
$6z=258$
$6z/6=258/6$
$z=43$
$y=5z-22$
$y=5*43-22$
$y=215-22$
$y=193$
$x=y-60$
$x=193-60$
$x=133$
$133+2*193+3*43$
$133+386+129$
$519+129$
$648$
